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Absence of Renormalization Group Pathologies Near the Critical Temperature - Two Examples (1996)
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BibTeX
@MISC{Haller96absenceof,
author = {Karl Haller and Tom Kennedy},
title = {Absence of Renormalization Group Pathologies Near the Critical Temperature - Two Examples},
year = {1996}
}
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Abstract
: We consider real space renormalization group transformations for Ising type systems which are formally defined by e \GammaH 0 (oe 0 ) = X oe T (oe; oe 0 )e \GammaH(oe) where T (oe; oe 0 ) is a probability kernel, i.e., P oe 0 T (oe; oe 0 ) = 1, for every configuration oe. For each choice of the block spin configuration oe 0 , let ¯ oe 0 be the measure on spin configurations oe which is formally given by taking the probability of oe to be proportional to T (oe; oe 0 )e \GammaH(oe) . We give a condition which is sufficient to imply that the renormalized Hamiltonian H 0 is defined. Roughly speaking, the condition is that the collection of measures ¯ oe 0 are in the high temperature phase uniformly in the block spin configuration oe 0 . The proof of this result uses methods of Olivieri and Picco. We use our theorem to prove that the first iteration of the renormalization group transformation is defined in the following two examples: decimation with spaci...
Keyphrases
critical temperature renormalization group pathology block spin configuration renormalization group transformation real space renormalization group transformation renormalized hamiltonian spin configuration roughly speaking high temperature phase configuration oe type system probability kernel first iteration
